You can check ADS connection in SAP via multiple different methods. Make sure before checking below ADS RFC should working fine in SM59.

1. SA38 – FP_CHECK_DESTINATION_SERVICE (With tick mark).

It should give your reply with tick mark.

2. SA38 – FP_PDF_TEST_00

3. SA38 – FP_TEST_00

If there are any other error like SOAP/exception or any, you need to check the Java server Node or check server Trace. if restart of server node works & ADS start working. 1 cause of this, it means you are running out of memory in system. check the memory at OS level.